Transversal action “ Models “ Phase 3 : Underground structure modelling - Gallery type 2

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B. Pardoen - F. Collin - S. Levasseur - R. Charlier

Université de Liège – ArGEnCo

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Outline

1. CONSTITUTIVE MODELS

2. MODELS FITTING

3. NUMERICAL MODELLING – TYPE 1

4. NUMERICAL MODELLING – FRACTURES MODELLING – TYPE 1 AND 2

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1. CONSTITUTIVE MODELS

2. MODELS FITTING

5. CONCLUSIONS AND OUTLOOKS

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1.1 Balance equations (LAGAMINE FE code) : Momentum :

Bishop’s effective stress : Water mass :

1.2 Flow model :

Advection of liquid phase (Darcy’s flow) :

Water retention and permeability curves (Van Genuchten’s model) :

1. Constitutive models

( _ij) 0 div   ij ij b S pij rw w ij    



w Sr w,



div



w q_w



0 t        , ( , ) ij r w r w w w w k k S q p    





, max 1 m n c r w res res r p S S S S P   _ _       _{  }      





2 1/ , , 1 1 , m m r w r w r w k  S _  S _  

Symbol Name Value Unit khor,// Horizontal intrinsic water permeability 4 10-20 m²

kvert,┴ Vertical intrinsic water permeability 1.33 10-20 m²

Φ Porosity 0.173 -

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1.3 Mechanical model : Elasto-viscoplasticity :

1.3.1 Linear elasticity theory : Stress-strain relation, Hooke law :

Orthotropic (9 param.) : Cross-anisotropic (5 param.) : Isotropic (2 param.) :

e p vp ij ij ij ij     ' ' , , ( ) e e e e e e

ij ijkl kl ij ijkl kl ijkl ijkl

d D d d C d C inv D 1 2 3 12 13 23 12 13 23 31 21 1 2 3 32 12 1 2 3 13 23 1 2 3 12 13 23 31 13 23 32 21 12 2 1 3 1 2 3 , , , , , , , , 1 - -1 - -1 - -1 2 1 2 1 2 , , e ijkl E E E G G G E E E E E E E E E D G G G E E E E E E                                                                                      / / / / / / // // / / / / / / / / / / / / / / / / / / / / // // / / / / / / / / / / // // / / // / / , , , , 1 - -1 - -1 - -1 1 2 1 2 e ijkl E E G E E E E E E E E E D E G G E E                       _       _                    _                       _                      , 1 - -- 1 -- - 1 1 1 1 1 e ijkl E D E                                                

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1.3.2 Plasticity theory :

- Non-associated, elasto-viscoplastic, internal friction model. Van Eeckelen yield surface :

- Hardening/softening of φ/c as a function of the Von Mises equivalent plastic strain :

- Cohesion anisotropy by microstructure fabric tensor :

For cross-anisotropy :

1. Constitutive models



 





0



0 . 2 ˆ ˆ : 3 p Cf C eq p p p p eq ij ij eq C C _p eq dec if dec B dec                       ˆ tan _C c F II_ m I_      _  _   3 0 , i i i ij i j i ij ij c a l l l          21 22 23



( ) ( ) ...



cc₀ 1A₁₁ 1 3 l₂2 B A_{1 11}2 1 3 l₂2 2

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1.3.3 Viscoplasticity theory :

- Loading surface of the viscoplastic flow : - Viscoplastic hardening function :

- Equivalent viscoplastic shear strain : from Zhou et al. (2008), Jia et al. (2008)

1.3.4 Biot’s coefficient anisotropy : Biot’s symmetric tensor :

Orthotropic (3 param.) : Cross-anisotropic (2 param.) : Isotropic (1 param.) :

ˆ 3 0 3 vp vp c s c I f II R A C R        _  _   





,0 1 ,0 vp vp vp vp vp vp B          2 ˆ ˆ 3 vp vp vp ij ij     3 e ijkk ij ij s C b K    11 22 33 ij b b b b            0 1 ij ij s b b b b b K b K      _ _       / / / / ij b b b b            

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Outline

2. MODELS FITTING

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2.1 Triaxial compression tests :

Isotropic model

Elastoplastic, undrained condition

Symbol Name Value Unit

E Young’s modulus 4000 MPa

ν Poisson’s ratio 0.3 -

b Biot’s coefficient 0.6 - ρs Specific mass 2750 kg/m³

ψ Dilatancy angle 0.5 °

φc0 Initial compression friction angle 10 ° φcf Final compression friction angle 23 ° Bφ Friction angle hardening coefficient 0.001 - decφ Friction angle hardening shifting 0 -

c Cohesion 4.2 MPa

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2.1 Triaxial compression tests :

Isotropic model

Elastoplastic, undrained condition

φc0 Initial compression friction angle 10 ° φcf Final compression friction angle 23 ° Bφ Friction angle hardening coefficient 0.001 - decφ Friction angle hardening shifting 0 -

c0 Initial cohesion 4.2 MPa cf Final cohesion 0.04-2 MPa Bc Cohesion softening coefficient 0.001 - decc Cohesion softening shifting 0.011 -

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2.2 Triaxial extension tests :

Isotropic model

Elastoplastic, undrained condition, p=cst, extension axiale

Drucker-Prager

Van Eeckelen

Lower resistance in extension

φc0 Initial compression friction angle 10 ° φcf Final compression friction angle 23 ° φe0 Initial extension friction angle 7 ° φef Final extension friction angle 23 ° Bφ Friction angle hardening coefficient 0.001 - decφ Friction angle hardening shifting 0 -

c0 Initial cohesion 4.2 MPa

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2.3 Creep tests :

Isotropic model

Elasto-viscoplastic, drained condition f_vp,0=0

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2.4 Anisotropic parameters :

- Cross-anisotropic elasticity : E_// = 5 [GPa] , E_┴ = 4 [GPa] , ν_//// = ν_//┴ = 0.3 , G_//┴ = 1.92 [GPa]

- Biot : b_// = 0.60 , b_┴= 0.66

- Cohesion : c₀=4.1 [MPa] , A₁₁=0.117, B₁=14.24

Cross-anisotropy Uniaxial compression

HR=96%, Srw=94% (Unit C), b=0.6, ϕ =23°

1 _{Projet HAVL – Argile « Expertise sur les mesures sur échantillons d'argilite du module de}

déformation et de la résistance à la compression simple », Ref.: C.RP.0ULg.08.001, 2008

Constitutive models Models fitting Numerical modelling Fractures modelling Conclusions 13

Orientation [°] 0° (┴) 30° 90° (//) R_c,mean [MPa] 1 _23.5 _18.7 _20.6 c’ [MPa] 6.3 4.8 5.4 1 sin ' ' tan ' 2sin ' c rw w c  R bS p      _  _  



( ) ( ) ...



cc₀ 1A₁₁1 3 l₂2 B A_{1 11}2 1 3 l₂2 2 cos( ) l₂  

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Anisotropic model

Triaxial compression tests Average values

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Anisotropic model

Triaxial compression test Average values

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Outline

2. MODELS FITTING

3. NUMERICAL MODELLING – TYPE 1

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3.1. Mesh / Boundary conditions / Initial conditions : - Mesh :

By symmetry: quarter of the gallery. Modelling in 2D plane strain state. Gallery radius = 2.6 m.

- Initial anisotropic stress state : p_w,0 = 4.7 [Mpa]

σ_h = σ_z = 12.40 [MPa] σ_v = σ_y = 12.70 [MPa]

σ_H = σ_x = 1.3 σ_h = 16.12 [MPa] - Hydraulic permeability anisotropy

k_hor/vert = 4 10-20_{/ 1.33 10}-20 _[m²] - Excavation : 0 0.2 0.4 0.6 0.8 1 0 4 8 12 16 20 24 28 Ta u x Time [days] l 1l 0 2 4 6 0 4 8 12 16 20 24 28 pw [M P a] Time [days]

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3.2. Isotropic model :

3.2.1 Hydromechanical modelling, case 1.3 :

- Convergence + influence of viscosity :

(a) no viscosity

No evolution of deformation in the long term. σ_H>σ_v , ε_H>ε_v

(b) viscosity

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- Plastic zone + influence of viscosity :

(a) no viscosity

HM coupling effect

(b) viscosity

HM coupling + viscosity effects

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- Pore water pressure :

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3.2.2 Mechanical modelling, case 1.1 and 1.2 :

- Convergence + influence of the support : (a) Case 1.1, flexible liner

(b) Case 1.2, rigid liner

Stops the convergence after the excavation.

(c) Case 1.3, flexible liner, HM modelling

Similar to mechanical modelling in the long term. Vertical convergence slightly greater.

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3. Numerical modelling – Galery type 1

3.2.2 Mechanical modelling, case 1.1 and 1.2 : - Radial and orthoradial total stress :

Vertical cross-section

(a) Case 1.1, flexible liner (b) Case 1.2, rigid liner (c) Case 1.3, flexible liner

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3.2.2 Mechanical modelling, case 1.1 and 1.2 : - Radial and orthoradial total stress :

Vertical cross-section

(a) Case 1.1, flexible liner (b) Case 1.2, rigid liner (c) Case 1.3, flexible liner

M modelling M modelling HM modelling

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3.3. Anisotropic model :

- Convergence + influence of the support : (a) Case 1.1, flexible liner, M modelling σ_H> σ_v , ε_H> ε_v

E_//,hor > E_┴,vert , ε_H< ε_v

(b) Case 1.2, rigid liner, M modelling

Stops the convergence after the excavation.

(c) Case 1.3, flexible liner, HM modelling

Vertical convergence is greater than horizontal one. This is due to the coupling !

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3.3. Anisotropic model :

- Plastic zone + influence of the support : (a) Case 1.1, flexible liner, M modelling Viscosity effects

(b) Case 1.2, rigid liner, M modelling Viscosity effects

(c) Case 1.3, flexible liner, HM modelling HM coupling + viscosity effects

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Outline

2. MODELS FITTING

4. NUMERICAL MODELLING – FRACTURES MODELLING – TYPE 1 AND 2

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4.1 Fracture modelling – strain localisation – coupled 2d_{gradient model :}_{(Chambon et al., 1998 and 2001)}

The continuum is enriched with microstructure effects. The kinematics include the classical one (macro) and the microkinematics (Toupin 1962, Mindlin 1964, Germain 1973).

Biphasic porous media : solid + fluid (Collin et al., 2006)

Balance equations :

S_ijk is the double stress, which needs an additional constitutive law = linear elastic law (Mindlin, 1964) function of the (micro) second gradient of displacement field :

It depends only on one elastic parameter D. The internal length scale is related to this parameter. (Chambon et al., 1998, Kotronis et al., 2007)

2 ijk

,

i j k

u

D

x x



_



 

_



_

 









* 2 * * * * * * * * q i i ij ijk i i i i i w i j j i w w i k w

p

M

u

d

G u d

t

p

m

d

Q p

u

T

d

q p d

t

x

Du

d

x

_



     





_





_{ }



_



 S

 

 













 



 





_













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60% excavation 80% excavation End of excavation 100 days 1000 days σ/σ₀ = 0.40 σ/σ₀ = 0.20 σ/σ₀ = 0.00 σ/σ₀ = 0.00 σ/σ₀ = 0.00

4.2 Galery type 1 - Hydromechanical modelling : - Isotropic model :

4. Numerical modelling – Fractures modelling

Total deviatoric strain Plasticity 2 ˆ ˆ 3 eq ij ij     0 0.184

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4.2 Galery type 1 - Hydromechanical modelling : - Isotropic model :

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4.2 Galery type 1 - Hydromechanical modelling : - Anisotropic model :

4. Numerical modelling – Fractures modelling

Total deviatoric strain

19.6 days 25 days 3 months

σ/σ₀ = 0.40 σ/σ₀ = 0.20 σ/σ₀ = 0.00 σ/σ₀ = 0.00 σ/σ₀ = 0.00 Plasticity 2 ˆ ˆ 3 eq ij ij     0 0.065 0 0.2 0.4 0.6 0.8 1 0 4 8 12 16 20 24 28 Ta u x Time [days] l 1l

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4.3 Galery type 2 - Hydromechanical modelling : - Anisotropic model :

19.6 days 25 days 3 months 100 years

σ/σ₀ = 0.40 σ/σ₀ = 0.20 σ/σ₀ = 0.00 σ/σ₀ = 0.00 0 0.2 0.4 0.6 0.8 0 4 8 12 16 20 24 28 Ta u x Time [days] l 1l

Constitutive models Models fitting Numerical modelling Fractures modelling Conclusions 31

Total deviatoric strain Plasticity 2 ˆ ˆ 3 eq ij ij     0 0.030 Deviatoric strain increment eq d

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4.3 Galery type 2 - Hydromechanical modelling :

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4.3 Galery type 1 and 2 - Hydromechanical modelling :

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Outline

2. MODELS FITTING

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